The Hausdorff dimension of multivariate operator-self-similar Gaussian random fields

Let {X(t) : t is an element of R-d} be a multivariate operator-self-similar random field with values in R-m. Such fields were introduced in [22] and satisfy the scaling property {X(c(E)t) : t is an element of R-d} =(d) {c(D)X(t) : t is an element of R-d} for all c > 0, where E is a d x d real matrix and D is an m x m real matrix. We solve an open problem in [22] by calculating the Hausdorff dimension of the range and graph of a trajectory over the unit cube K = [0, 1](d) in the Gaussian case. In particular, we enlighten the property that the Hausdorff dimension is determined by the real parts of the eigenvalues of E and D as well as the multiplicity of the eigenvalues of E and D. (C) 2017 Elsevier B.V. All rights reserved.